Homeomorphism examples

A homotopy between closed curves $gamma_1$ and $gamma_2$ is defined as a continuous map $H:[0,1]times[0,1] to X$ that satisfies $H(0,t) = gamma_1(t)$ and $H(1,t) = gamma_2(t)$ for all $tin [0,1]$. Since homotopies involve continuous deformations, the closed curves $fcircgamma_1$ and $fcircgamma_2$ in $Y$ are homotopic if $f: X to Y$ is a homeomorphism and $gamma_1$, $gamma_2$ are homotopic in $X$. This information can be used to define homeomorphism equivalently as a bijective, continuous, open map that maps open sets to open sets.


Question:

What’s a homeomorphism?

A function, denoted by $fcolon mathbb {N} to mathbb {Q}$, maps any natural number to a rational number.

Function $f$ is defined as $fcolon mathbb {R} to mathbb {R}$, where the mapping is such that $x$ is mapped to $x^3$.

Define $f$ as a function that maps the interval $[0, 2pi)$ to the set $S^1$, where $S^1$ represents the set of complex numbers $z$ such that $|z|=1$. This mapping is defined as $varphi$ being mapped to $e^{ivarphi}$.

I’m confident that b is non-functional since it lacks bijectivity, and I believe a is functioning correctly. However, I’m uncertain about c and would appreciate an explanation from someone.


Solution 1:

Bijections between $mathbb{N}$ and $mathbb{Q}$ exist, but none of them qualify as a homeomorphism. Specifically, if $x in mathbb{N}$, then ${x}$ is an open set in $mathbb{N}$, while $f({x}) = {f(x)}$ is never open in $mathbb{Q}$.

The function is a homeomorphism and its continuity is guaranteed by its inverse, which is given by $f^{-1}(x) = sqrt[3]{x}$.

The interval $[0,2pi)$ is not a compact set while $S^1$ is compact. Therefore, they cannot be considered as equivalent.


Solution 2:


Assume that $X$ and $Y$ are topological spaces, and let $fcolon Bbb Rto Bbb R$ be a bijective function. If both $f$ and $f^{-1}$ are continuous, then $f$ is a homeomorphism. With that in mind, let’s return to the previous examples.

Before stating that $f$ is a homeomorphism, it is necessary to define the function $f: Bbb N to Bbb Q$.

The function $f(x)=x^3$ is a bijection and both $f$ and its inverse function $f^{-1}$ are continuous, making $f$ a homeomorphism.

Please verify $c$ and let me know if that resolves the issue.

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